The Random Normal Scores Test (RNST) has been suggested as a powerful alternative to the use of the parametric analysis of variance (ANOVA) test when the underlying population is non-normally distributed. The major support for this suggestion rests on asymptotic theory. An empirical analysis of the RNST performed under the F and Chi-square distributions is reported. A descriptive exploratory design was used. Monte Carlo methods were used to generate the data and to describe their characteristics. Two sample sizes were investigated. For the smaller sample size of 2 observations per cell, 42 treatment combinations of main effects and interactions were investigated. For the larger sample size of 20 observations per cell, 26 treatment combinations were studied. Results were charted in 390 tables; 7 of these tables are included as a representative sample. The RNST was shown to be non-robust and not a powerful alternative to the ANOVA test in the balanced 2x2x2 layout for various population distributions, sample sizes, and nominal alpha levels selected. A suitable alternative to ANOVA under conditions of non-normal distributions is still required. (SLD)